In her original remarks, Baroness Gardner said: “The housing issue is not such a battle now. It has been addressed by putting a ceiling on housing benefit.

“I heard the noble Lord, Lord Wigley, talk about Wales and the shortage of work there.

“Surely a lot of those people who cannot get anywhere else to live in the country could go to Wales.

“If you are living on a benefit, it does not matter where you are living if it is fully paid for by someone else.

“Wales has always had a marvellous reputation for literacy, and I am quite sure that the schools would be good there. You want to be able to live in a safe, clean environment.

“Again, Wales is a beautiful country. I am a New South Welshman myself.”

Plaid Cymru’s Lord Wigley, who spoke prior to Baroness Gardner in the debate, later said he was concerned by any potential plan to “cleanse” Greater London.

Lord Wigley said: “I was quite surprised, as were a number of colleagues from various groups across the benches that she should have said that.

“The implication is that a young mother with children in school can’t afford to keep up their payments with rents very high in London and that they would move to Merthyr Tydfil or Middlesbrough.

“Well, goodness me, it would be moving them away from all their family and support network but also moving them into areas of high unemployment.

“In areas like Blaenau Gwent and the Rhondda there are 11 or 12 people chasing one job.”

Lord Wigley added it was “unacceptable” if public policy went in that direction.

Defending her remarks, Baroness Gardner told Wales on Sunday she was not pinpointing Wales in particular but was making a general point that if people were desperate living in London they could move elsewhere.

She said: “It was not Wales in particular but someone in Wales had just spoken.

“I was saying that you could have a good life in any of those places, that if you were living on a benefit that really you weren’t obliged to stay in the most expensive parts of the country.

“I uprooted myself and moved to this country. It was sheer chance I lived in London because that’s where all the Australians went to.”

Meanwhile Baroness Gardner, who was formerly on Westminster City Council with responsibility for an eighth of London’s housing, said she was “very impressed” with Cardiff when she had visited but said the biggest difference between South Wales and New South Wales was the weather.

“It was called that by Captain Cook because he thought it reminded him of South Wales,” she said.

“I think the climate is a very big factor. The fact the climate is very good and across Australia is booming.”

A Welsh Government spokesman said it was wrong of the Baroness to suggest it made no difference to a household on benefits where they lived.

“We are concerned about the impact of the benefit cap and housing benefit reform,” he said.

“We are already in the midst of hugely challenging times in terms of meeting the housing needs of Welsh people.

“It is wrong to suggest that it makes no difference to a household in receipt of benefits where they are living.

“It makes a difference to the families who have connections to their local community and it makes a difference to the demands on local public services.”